Binary LCD Codes from $\mathbb{Z}_2\mathbb{Z}_2[u]$

TL;DR

This paper extends the concept of LCD codes to a new ring structure, characterizes their properties, and constructs improved binary LCD codes from these ring-based codes.

Contribution

It introduces $ ext{Z}_2 ext{Z}_2[u]$-LCD codes, characterizes their structure, and demonstrates how to generate binary LCD codes with better parameters.

Findings

Binary images of $ ext{Z}_2 ext{Z}_2[u]$-LCD codes are binary LCD codes.

New binary LCD codes with improved parameters are constructed.

The paper generalizes LCD codes to a novel ring setting.

Abstract

Linear complementary dual (LCD) codes over finite fields are linear codes satisfying $C\cap C^{\perp}=\{0\}$. We generalize the LCD codes over finite fields to $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes over the ring $\mathbf{Z}_2\times(\mathbf{Z}_2+u\mathbf{Z}_2)$. Under suitable conditions, $\mathbb{Z}_2\mathbb{Z}_2[u]$-linear codes that are $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes are characterized. We then prove that the binary image of a $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD code is a binary LCD code. Finally, by means of these conditions, we construct new binary LCD codes using $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes, most of which have better parameters than current binary LCD codes available.

Tables

Table 1. Table 1 : New binary LCD codes obtained from Corollary 3.7

self-orthogonal codes in Ref [2]	LCD codes in Ref[11],[18],[8],[16]	New binary LCD codes
${[18,2,8]}_2$	${[10,2,6]}_2$,	${[28,6,\ge 14]}_2$
	${[12,2,7]}_2$,	$[30,2,\ge 15]]{}_2$,
	${[13,2,8]}_2$,	${[31,2,\ge 16]}_2$,
	${[14,2,9]}_2$	$[32,2,\ge 17]]{}_2$,
	${[15,2,10]}_2$,	${[33,2,\ge 18]}_2$
${[56,25,6]}_2$	${[35,25,4]}_2$	${[91,25,\ge 10]}_2$
${[56,28,6]}_2$	${[35,28,\ge 3]}_2$	${[91,28,\ge 9]}_2$
	${[65,28,14]}_2$,	${[121,28,\ge 20]}_2$
	${[66,28,14]}_2$,	$[122,28,\ge 20]]{}_2$
${[98,48,6]}_2$	${[56,48,\ge 3]}_2$,	${[154,48,\ge 9]}_2$
	${[63,48,6]}_2$,	${[161,48,\ge 12]}_2$
	${[257,48,74]}_2$,	${[355,48,\ge 80]}_2$

Equations

$${\mathbb{Z}}_2\times R=\{(a,b)\mid a\in {\mathbb{Z}}_2\mathrm{and}b\in R\}.$$

$$r(a,b)=(\theta (r)a,rb).$$

$$r(a_1,\dots ,a_{\alpha },b_1,\dots ,b_{\beta })=(\theta (r)a_1,\dots ,\theta (r)a_{\alpha },r{b}_1,\dots ,r{b}_{\beta }).$$

$$\Phi :{\mathbb{Z}}_2^{\alpha }\times R^{\beta }⟶{\mathbb{Z}}_2^n,$$

$$\Phi (a_1,\dots ,a_{\alpha },s_1+u{t}_1,\dots ,s_{\beta }+u{t}_{\beta })=(a_1,\dots ,a_{\alpha },t_1,\dots ,t_{\beta },s_1+t_1,\dots ,s_{\beta }+t_{\beta }).$$

$$W_{L}(r)=\left\{\begin{array}[]{ll}0,&\hbox{if\quad$r=0$;}\\ 1,&\hbox{if\quad$r=1,1+u$;}\\ 2,&\hbox{if\quad$r=u$.}\end{array}\right.$$

$$[\mathbf{v},\mathbf{w}]=u\sum _{i=1}^{\alpha }{a}_i{b}_i+\sum _{j=1}^{\beta }{x}_j{y}_j\in R.$$

$$C^{\perp }=\{\mathbf{v}\in {\mathbb{Z}}_2^{\alpha }\times R^{\beta }\mid [\mathbf{v},\mathbf{w}]=0\mathrm{for}\mathrm{all}\mathbf{w}\in C\}.$$

$$G=\left(\begin{array}[]{ccccc}I_{k_{0}}&A_{1}&0&0&uP\\ 0&S&I_{k_{1}}&A&B_{1}+uB_{2}\\ 0&0&0&uI_{k_{2}}&uD\\ \end{array}\right),$$

$$H=\left(\begin{array}[]{ccccc}A_{1}^{T}&I_{\alpha-k_{0}}&-uS^{T}&0&0\\ -P^{T}&0&-(B_{1}+uB_{2})^{T}+D^{T}A^{T}&-D^{T}&I_{\beta-k_{1}-k_{2}}\\ 0&0&-uA^{T}&uI_{k_{2}}&0\\ \end{array}\right).$$

$$[\mathbf{x},\mathbf{y}]=[\mathbf{b},\mathbf{e}]+u([\mathbf{a},\mathbf{d}]+[\mathbf{b},\mathbf{f}]+[\mathbf{c},\mathbf{e}]),$$

$$[\Phi (\mathbf{x}),\Phi (\mathbf{y})]=[\mathbf{a},\mathbf{d}]+[\mathbf{c},\mathbf{f}]+[\mathbf{b},\mathbf{e}]+[\mathbf{b},\mathbf{f}]+[\mathbf{c},\mathbf{e}]+[\mathbf{c},\mathbf{f}]=[\mathbf{b},\mathbf{e}]+([\mathbf{a},\mathbf{d}]+[\mathbf{b},\mathbf{f}]+[\mathbf{c},\mathbf{e}]).$$

$$\Phi (C^{\perp })\subset \Phi (C{)}^{\perp }.$$

$$|\Phi (C^{\perp })|=|\Phi (C{)}^{\perp }|$$

$${\mathbf{w}}_s=\sum _{j=1}^{s-1}{\alpha }_j{\mathbf{w}}_j.$$

$${\alpha }_1{\mathbf{w}}_1+\cdots +{\alpha }_{s-1}{\mathbf{w}}_{s-1}+(-1){\mathbf{w}}_s=0,$$

$$[\sum _{j=1}^n{x}_{j}G(j,:),G(i,:)]=0\mathrm{for}\mathrm{any}i\in \{1,2,\dots ,k\},$$

$$G{G}^T\mathbf{x}=0,$$

$$|G{G}^T|=|I_k+A{A}^T|=|(-1)((-1)I_k-A{A}^T)|=(-1{)}^k|(-1)I_k-A{A}^T)|\ne 0.$$

$$G_{2}=\left(\begin{array}[]{ccccc}I_{k_{0}}&A_{1}&0&0&uP\\ 0&S&I_{k_{1}}&A&B_{1}+uB_{2}\\ 0&0&0&uI_{k_{2}}&uD\\ \end{array}\right),$$

$$G=\left(\begin{array}[]{ccccc}1&0&0&u&u\\ 0&1&1&u&u\end{array}\right).$$

$$\begin{array}{rl}\mathbf{v}& =a_1(1,0,0,u,u)+a_2(0,1,1,u,u)=(\theta (a_1),\theta (a_2),a_2,a_{1}u,(a_1+a_2)u).\end{array}$$

$$\{\begin{array}{r}{b}_2^2=0,\\ \theta (b_1{)}^2+\theta (b_2{)}^2=0,\\ \theta (b_1)=0,\\ u\theta (b_2)+b_2=0.\end{array}$$

$${\delta }_{1}G(1,:)+\cdots +{\delta }_{k}G(k,:)=0.$$

$$C_{\alpha }=\{\mathbf{a}\in {\mathbb{Z}}_2^{\alpha }|\mathrm{there}\mathrm{exist}\mathbf{b}\in R^{\beta }\mathrm{such}\mathrm{that}(\mathbf{a},\mathbf{b})\in C\},$$

$$C_{\beta }=\{\mathbf{b}\in R^{\beta }|\mathrm{there}\mathrm{exist}\mathbf{a}\in {\mathbb{Z}}_2^{\alpha }\mathrm{such}\mathrm{that}(\mathbf{a},\mathbf{b})\in C\}.$$

$$G=\left(\begin{array}[]{cccccc}1&0&0&1&0&0\\ 0&1&0&0&1&0\\ 0&0&1&0&0&1\\ 1&1&1&u&u&u\\ \end{array}\right).$$

$$\Phi (C\cap C^{\perp })\subset \Phi (C)\cap \Phi (C^{\perp }).(3.1)$$

$$\Phi (C\cap C^{\perp })\supset \Phi (C)\cap \Phi (C^{\perp }).(3.2)$$

$$\Phi (\mathbf{x})=(a_0,\dots ,a_{s-1},c_0,\dots ,c_{t-1},b_0+c_0,\dots ,b_{t-1}+c_{t-1})\in \Phi (C)\cap \Phi (C^{\perp }).$$

Full text

Key Words: $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes; $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD codes; self-orthogonal codes

**Mathematics Subject Classification (2010) **94B05 11T71

1 Introduction

Linear complementary dual codes (which is abbreviated to LCD codes) are linear codes that meet their dual trivially. These codes were introduced by Massey in [14] and showed that asymptotically good LCD codes exist, and provide an optimum linear coding solution for the two-user binary adder channel. They are also used in counter measure to passive and active side channel analyses on embedded cryto-systems(See[3]). In recently, Guenda, Jitman and Gulliver investigated an application of LCD codes in constructed good entanglement-assisted quantum error correcting codes [9].

Yang and Massy in [19] showed that a necessary and sufficient condition for a cyclic code of length $n$ over finite fields to be an LCD code is that the generator polynomial $g(x)$ is self-reciprocal and all the monic irreducible factors of $g(x)$ have the same multiplicity in $g(x)$ as in $x^{n}-1$. In [17], Sendrier indicated that linear codes with complementary-duals meet the asymptotic Gilbert-Varshamov bound. Dougbherty, Kim, Ozkaya, Sok and Sol$\acute{e}$ developed a linear programming bound on the largest size of an LCD code of given length and minimum distance [7]. Ding, C. Li and S. Li in [6] constructed LCD BCH codes. Liu, Fan and Liu [13] introduced the so-called $k$-Galois LCD codes which include the usual LCD codes and Hermitian LCD codes as two special cases, and gave sufficient and necessary conditions for a code to be a $k$-Galois LCD code. In recently, Carlet et al. solved the problem of the existence of $q$-ary LCD MDS codes for Euclidean case [4], they also introduced a general construction of LCD codes from any linear codes. Further more, they showed that any linear code over $\mathbb{F}_{q}$ ($q>3$) is equivalent to an Euclidean LCD code and any linear code over $\mathbb{F}_{q^{2}}$ ($q>2$) is equivalent to a Hermitian LCD code [5]. But, in [15], B. Pang, S. Zhu, and X. Kai show that LCD codes are not equivalent to linear codes over $\mathbb{F}_{2}$. This motivates us to study binary LCD codes.

We finish this introduction with a description of each section in this paper. Section 2 reviews the basics about $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes and LCD codes. In Section 3, we propose new constructions of binary LCD codes by using $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD codes, and, in Section 4, using methods of the section 3, concrete examples are presented to construct good parameters binary LCD codes. Finally, a brief summary of this work is described in Section 5.

2 Preliminaries

In order for the exposition in this paper to be self-contained, we introduce some basic concepts and results about $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes and LCD codes. For more details, we refer to [1],[14],[12].

Starting from this section till the end of this paper, we denote the ring $\mathbb{Z}_{2}+u\mathbb{Z}_{2}$ by $R$, where $u^{2}=0$. It is easy to see that the ring $\mathbb{Z}_{2}$ is a subring of the ring $R$. We define the set

[TABLE]

Let $r=s+ut\in R$. Define the map $\theta:R\longrightarrow\mathbb{Z}_{2}$, such that $\theta(r)=s$. So, $\theta(0)=0,\theta(1)=1,\theta(u)=0$, and $\theta(1+u)=1$. Obviously, the mapping $\theta$ is a homomorphism from ring $R$ to $\mathbb{Z}_{2}$. Now, for any element $r\in R$, define an $R$-scalar multiplication on $\mathbb{Z}_{2}\times R$ as

[TABLE]

Furthermore, this multiplication can be extended naturally to $\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$ as follows. For any $r\in R$ and $\mathbf{v}=(a_{1},\ldots,a_{\alpha},b_{1},\ldots,b_{\beta})\in\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$ define

[TABLE]

Definition 2.1**.**

A $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ is a non-empty $R$-submodule of $\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$. If $C\subset\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code, group isomorphic to $\mathbb{Z}_{2}^{k_{0}+k_{1}}\times\mathbb{Z}_{2}^{k_{2}}$, then C is called a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code of type $(\alpha,\beta,k_{0},k_{1},k_{2})$ where $k_{0},k_{1}$ and $k_{2}$ are defined above.

Now we recall the definition of the Gray map on $\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$. Observe that any element $r\in R$ can be expressed as $r=s+ut$, where $r,q\in\mathbb{Z}_{2}$ . The Gray map on $\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$, defined in [1], can be written as

[TABLE]

[TABLE]

Note that the binary image $\Phi(C)$ of a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ of type $(\alpha,\beta,k_{0},k_{1},k_{2})$ is a binary linear code of length $n=\alpha+2\beta$ and size $2^{k_{0}+2k_{1}+k_{2}}$. It is also called a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code.

The Lee weight $W_{L}(r)$ of the element $r\in R$ is defined as

[TABLE]

Denote by $W_{L}(\mathbf{b})$ the Lee weight of $\mathbf{b}\in R^{\beta}$, which is the rational sum of Lee weights of the coordinates of $\mathbf{b}$. For a vector $\mathbf{v}=(\mathbf{a}|\mathbf{b})\in\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$, the Lee weight of $\mathbf{v}$ is defined as $W_{L}(\mathbf{v})=W_{H}(\mathbf{a})+W_{L}(\mathbf{b})$, where $W_{H}(\mathbf{a})$ is the Hamming weight of $\mathbf{a}\in\mathbb{Z}_{2}^{\alpha}$. The minimum Lee distance $d_{L}(C)$ of a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ is the smallest nonzero Lee distance between all pairs of distinct codewords of $C$. Obviously, $d_{L}(C)=W_{H}(\Phi(C))$.

The inner product in $\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$, defined in [1], can be written as

[TABLE]

where $\mathbf{v}=(a_{1},a_{2},\ldots,a_{\alpha},x_{1},\ldots,x_{\beta})$ and $\mathbf{w}=(b_{1},b_{2},\ldots,b_{\alpha},y_{1},\ldots,y_{\beta})$ .

The dual code of a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ is defined in the standard way by

[TABLE]

We say that the code $C$ is self-orthogonal, if $C\subset C^{\perp}$ and self-dual, if $C=C^{\perp}$.

A $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ is called $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD if $C^{\perp}\cap C=\{\mathbf{0}\}$.

Lemma 2.2**.**

If $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code of type $(\alpha,\beta,k_{0},k_{1},k_{2})$, then $|C|\cdot|C^{\perp}|=2^{\alpha+2\beta}$.

**Proof.  **According to [2, Theorem 2.2 and Theorem 2.7], $C$ is permutation equivalent to a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code with the standard form matrix

[TABLE]

and, its dual code $C^{\perp}$ with the generator matrix

[TABLE]

Thus, $|C|\cdot|C^{\perp}|=2^{k_{0}+2k_{1}+k_{2}}2^{\alpha-k_{0}+2(\beta-k_{1}-k_{2})+k_{2}}=2^{\alpha+2\beta}$.

Theorem 2.3**.**

Let $C$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code of type $(\alpha,\beta,k_{0},k_{1},k_{2})$. Then $\Phi(C^{\bot})=\Phi(C)^{\bot}$, where $\Phi(C)^{\bot}$ is the ordinary dual of $\Phi(C)$ as a binary code.

**Proof.  **Let $\mathbf{x}=(\mathbf{a},\mathbf{b}+u\mathbf{c}),\mathbf{y}=(\mathbf{d},\mathbf{e}+u\mathbf{f})\in C$ be two codewords, where $\mathbf{a,d}\in\mathbb{Z}_{2}^{\alpha}$, and $\mathbf{b,c,e,f}\in\mathbb{Z}_{2}^{\beta}$. Then

[TABLE]

and

[TABLE]

It is easy to check that $[\mathbf{x,y}]=0$ implies $[\Phi(\mathbf{x}),\Phi(\mathbf{y})]=0.$ Therefore,

[TABLE]

But, by the definition of $\Phi$, $\Phi(C)$ is a binary linear code of length $\alpha+2\beta$ of size $|C|$. So, by the usual properties of the dual of binary codes, we know that $|\Phi(C)^{\bot}|=\frac{2^{\alpha+2\beta}}{|C|}.$

On the other hand, by Lemma 2.2, we have $|\Phi(C^{\bot})|=|C^{\bot}|=\frac{2^{\alpha+2\beta}}{|C|}$.

Thus

[TABLE]

Combining (2.1) and (2.2), we get the desired equality. ∎

3 $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD Codes

It is easy to see that $R$ is a local Frobenius ring with unique maximal ideal $\mathfrak{m}=\{0,u\}$.

We begin with some definitions and lemmas with respect to vectors in $\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$.

Definition 3.1**.**

Let $\mathbf{w}_{1},\ldots,\mathbf{w}_{k}$ be non-zero vectors in $\mathbb{Z}_{2}^{\alpha}\times R^{\beta}$ . Then $\mathbf{w}_{1},\ldots,\mathbf{w}_{k}$ are $R$-independent if $\sum_{j=1}^{k}\delta_{j}\mathbf{w}_{j}=0$ implies that $\delta_{j}\mathbf{w}_{j}=0$ for all $j$, where $\delta_{j}\in R$.

Following Definition 3.1, we can easily get the following lemma.

Lemma 3.2**.**

If the non-zero vectors $\mathbf{w}_{1},\ldots,\mathbf{w}_{s}$ in $\mathbb{Z}^{\alpha}\times R^{\beta}$ are $R$-independent and $\sum_{j=1}^{s}\delta_{j}\mathbf{w}_{j}=0$, then $\delta_{j}\in\mathfrak{m}$ for all $j$.

**Proof.  **Since $\sum_{j=1}^{s}\delta_{j}\mathbf{w}_{j}=0$. Then $\delta_{j}\mathbf{w}_{j}=0$ for all $j$. If $\delta_{j}\notin\mathfrak{m}$ for some $j$, then $\delta_{j}$ is a unit, and this implies that $\mathbf{w}_{j}=0$, which is a contradiction. ∎

Let $\mathbf{w}_{1},\ldots,\mathbf{w}_{s}$ be vectors in $\mathbb{Z}^{\alpha}\times R^{\beta}$. As usual, we denote the set of all linear combinations of $\mathbf{w}_{1},\ldots,\mathbf{w}_{s}$ by $\langle\mathbf{w}_{1},\ldots,\mathbf{w}_{s}\rangle$.

Lemma 3.3**.**

If the non-zero vectors $\mathbf{w}_{1},\ldots,\mathbf{w}_{s}$ in $\mathbb{Z}^{\alpha}\times R^{\beta}$ are $R$-independent, then $\mathbf{w}_{1},\ldots,\mathbf{w}_{s}$ none is a linear combination of the other vectors.

**Proof.  **Without loss of generality, suppose $\mathbf{w}_{s}$ can be written as a linear combination of the other vectors. Then

[TABLE]

This gives

[TABLE]

which is a contradiction to Lemma 3.2.

Central to the study of algebraic coding theory is the concept of a code generator matrix. The rows of the generator matrix form a basis of the code. We shall now give a definition of a basis of a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code.

Definition 3.4**.**

Let $C\neq\{\mathbf{0}\}$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code. The non-zero codewords $\mathbf{c}_{1},\mathbf{c}_{2},\ldots,\mathbf{c}_{k}$ are called a basis of $C$ if they are $R$-independent and generate $C$. Set $G=\begin{pmatrix}\mathbf{c}_{1}\\ \mathbf{c}_{2}\\ \vdots\\ \mathbf{c}_{k}\end{pmatrix}$. We say that $G$ is a generator matrix of $C$.

Given a $k\times n$ matrix $G$, we denote by $G(i,:)$ the $i$-th row of $G$.

In terms of the generator matrix, we now give a sufficient condition for a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code to be LCD.

Theorem 3.5**.**

Let $C\neq\{\mathbf{0}\}$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code with generator matrix $G$. If the $k\times k$ matrix $GG^{T}$ is invertible, then $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code, where $k$ is the number of rows of $G$.

**Proof.  **Suppose $\mathbf{a}\in C\cap C^{\perp}$. Then, by $\mathbf{a}\in C$, there exist $x_{1},\ldots,x_{k}\in R$ such that $\mathbf{a}=\sum_{j=1}^{n}x_{j}G(j,:)$, where $G(j,:)$ is $j$-th row of $G$.

Since $\mathbf{a}\in C^{\perp}$, we have

[TABLE]

that is

[TABLE]

where $\mathbf{x}=(x_{1},\ldots,x_{k})^{T}$. It follows that $\mathbf{x}=0$ since $GG^{T}$ is invertible. Hence $C\cap C^{\perp}=\{0\}$, i.e., $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. ∎

Based on this Theorem we get the following two corollaries.

Corollary 3.6**.**

Let $C$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code with generator matrix $G=\begin{pmatrix}I_{k}&A&0&uB\end{pmatrix}$. If $-1\notin\mathrm{Spec}(AA^{T})$, where $\mathrm{Spec}(M)$ denotes the of all eigenvalues of the matrix $M$, then $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code.

**Proof.  **Since $-1\notin\mathrm{Spec}(AA^{T})$, we have

[TABLE]

Hence, $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. ∎

Corollary 3.7**.**

Suppose that $C_{1}$ is an $[n_{1},k,d_{H}(C_{1})]$ binary LCD code with a generator matrix $G_{1}$ and $C_{2}$ a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$ self-orthogonal code of type $(\alpha,\beta,k_{0},k_{1},k_{2})$ with the standard form matrix

[TABLE]

where $k=k_{0}+k_{1}+k_{2}$. Then the code $C$ with a generator matrix $G=(G_{1}\mid G_{2})$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code, and $d_{L}(C)\geq d_{H}(C_{1})+d_{L}(C_{2})$.

**Proof.  **Clearly, we have $GG^{T}=G_{1}G_{1}^{T}$. Since $C_{1}$ is a binary LCD code, then $G_{1}G_{1}^{T}$ is invertible by [15, Proposition 1]. And so $GG^{T}$ is invertible. According to Theorem 3.5, $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. The minimum Lee distance of $C$ follows from the minimum Hamming distance of $C_{1}$ and the minimum Lee distance of $C_{2}$. ∎

Corollary 3.8**.**

Suppose that $C_{i}$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code with generator matrix $G_{i}$ for $i=1,2$. Then the code $C$ with generator matrix $G=G_{1}\otimes G_{2}$ is also a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code, where $G_{1}\otimes G_{2}$ denotes the Kronecker product of $G_{1}$ and $G_{2}$. Moreover, $d_{L}(C)=d_{L}(C_{1})d_{L}(C_{2})$.

**Proof.  **Note that $(G_{1}\otimes G_{2})(G_{1}\otimes G_{2})^{T}=(G_{1}G_{1}^{T})\otimes(G_{2}G_{2}^{T})$. Since $G_{i}G_{i}^{T}$ (for $i=1,2$) is invertible, we have $((G_{1}G_{1}^{T})\otimes(G_{2}G_{2}^{T}))((G_{1}G_{1}^{T})^{-1}\otimes(G_{2}G_{2}^{T})^{-1})=((G_{1}G_{1}^{T})(G_{1}G_{1}^{T})^{-1})\otimes(G_{2}G_{2}^{T})(G_{2}G_{2}^{T})^{-1})=I_{k_{1}}\otimes I_{k_{2}}=I_{k_{1}k_{2}}$, where $I_{a}$ is the identity matrix of order $a$. This means that $(G_{1}G_{1}^{T})\otimes(G_{2}G_{2}^{T})$ is invertible. By Theorem 3.5, $C_{1}\otimes C_{2}$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. ∎

The following example shows that the converse of Theorem 3.5, in general, does not hold.

Example 1**.**

Let $C$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code of type $(2,3,2,0,0)$ with generator matrix in standard form as follows

[TABLE]

Obviously, the determinant of $GG^{T}$ is equal to zero, i.e., $GG^{T}$ is not invertible. We now prove that $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. In fact, for any $\mathbf{v}\in C$, there are $a_{1},a_{2}\in\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$ such that

[TABLE]

Let $\mathbf{v}_{1}=(1,0,0,u,u),~{}\mathbf{v}_{2}=(0,1,1,u,u)$. Assume that $\mathbf{w}\in C\cap C^{\perp}$. Then by $\mathbf{w}\in C$ we can find elements $b_{1}$ and $b_{2}$ of $R$ such that $\mathbf{w}=(\theta(b_{1}),\theta(b_{2}),b_{2},b_{1}u,(b_{1}+b_{2})u).$ Since $\mathbf{w}\in C^{\perp}$, we have $[\mathbf{w},\mathbf{w}]=0$ and $[\mathbf{w},\mathbf{v}_{i}]=0$ for $1\leq i\leq 2$. Hence we have

[TABLE]

The implies that $b_{2}=0$, $b_{1}=0$ or $u$. Thus $\mathbf{w}=0$, i.e., $C\cap C^{\perp}=\{0\}$. We have shown that $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code.

Under suitable conditions, a converse to Theorem 3.5 holds.

Theorem 3.9**.**

Let $C\neq\{\mathbf{0}\}$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code with generator matrix $G$. For $1\leq i\leq k$, if every row $G(i,:)=(g_{i1},\ldots,g_{i,\alpha},\ldots,g_{i,\alpha+\beta})$ there exists $\alpha<j_{i}\leq\alpha+\beta$ such that $g_{i,j_{i}}$ is a unit of $R$. Then $C$ is $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD if and only if the $k\times k$ matrix $GG^{T}$ is invertible, where $k$ is the number of rows of $G$.

**Proof.  **The sufficient part follows from Theorem 3.5.

The following we prove the necessary condition. Suppose that $GG^{T}$ is not invertible. Then there is a nonzero vector $\mathbf{b}=(\delta_{1},\ldots,\delta_{k})\in R^{k}$ such that $GG^{T}\mathbf{b}^{T}=0$. Set $\mathbf{a}=\mathbf{b}G$. If $\mathbf{a}$ is a zero vector of $C$, then

[TABLE]

Since $G(1,:),\cdots,G(k,:)$ are $R$-independent, it follows that $\delta_{j}G(j,:)=0$ and $\delta_{j}\in\mathfrak{m}$ for all $j$. This is a contradiction to the assumption. Thus $\mathbf{a}\neq 0$.

By $GG^{T}\mathbf{b}^{T}=0$, we have also $G\mathbf{a}^{T}=0$, which is implies that $\mathbf{a}\in C^{\perp}$.

This gives $C\cap C^{\perp}\neq\{0\}$, i.e., $C$ is not a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. ∎

Let $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code. Define

[TABLE]

and

[TABLE]

It is easy to prove that $G_{\alpha}$ is the generator matrix of $C_{\alpha}$ and $G_{\beta}$ is the generator matrix of $C_{\beta}$, then $G=(G_{\alpha}|G_{\beta})$ is the generator matrix of the $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$.

Let $C$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code. If $C=C_{\alpha}\times C_{\beta}$ , then $C$ is called separable.

Let $C_{\beta}$ be a linear code of length $\beta$ over $R$. If $C_{\beta}\cap C_{\beta}^{\perp}=\mathbf{0}$, then $C_{\beta}$ is said to be an $R$-LCD code.

Theorem 3.10**.**

Let $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code. If $C_{\alpha}$ and $C_{\beta}$ are binary LCD and $R$-LCD codes,respectively, then $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code.

**Proof.  **Suppose that $C$ is not a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. Then exist $0\neq\mathbf{v}=(\mathbf{x},\mathbf{y})\in C\cap C^{\perp}$. We divide the rest of the proof into two cases.

Case 1. If $\mathbf{x}\neq 0$, then by $\mathbf{v}=(\mathbf{x},\mathbf{y})\in C\cap C^{\perp}$ we have $\mathbf{x}\in C_{\alpha}\cap C_{\alpha}^{\perp}$, which is a contradiction.

Case 2. If $\mathbf{y}\neq 0$, then by $\mathbf{v}=(\mathbf{x},\mathbf{y})\in C\cap C^{\perp}$ we have $\mathbf{y}\in C_{\beta}\cap C_{\beta}^{\perp}$, which is also a contradiction. ∎

Corollary 3.11**.**

Let $C=C_{\alpha}\times C_{\beta}$. Then $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code if and only if $C_{\alpha}$ and $C_{\beta}$ are binary LCD and $R$-LCD codes,respectively.

The reverse statements of Theorem 3.10 is not true in general.

Example 2**.**

Let $C$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code generated by

[TABLE]

Obviously, $C_{\alpha}=\mathbb{Z}_{2}^{\alpha}$ and $C_{\beta}=R^{\beta}$ are binary LCD and $R$-LCD codes, respectively. However, the last row $\mathbf{a}=(1~{}1~{}1~{}u~{}u~{}u)$ is orthogonal to any row in the generator matrix. Hence, $\mathbf{a}\in C\cap C^{\models}$ and C is not a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code.

Theorem 3.12**.**

A $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ is $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD if and only if $\Phi(C)$ is a binary LCD code.

**Proof.  **We firstly prove that $\Phi(C\cap C^{\perp})=\Phi(C)\cap\Phi(C^{\perp}).$

In fact, if $\Phi(\mathbf{w})\in\Phi(C\cap C^{\perp})$, then $\mathbf{w}\in C\cap C^{\perp}$. Thus, $\Phi(\mathbf{w})\in\Phi(C)$, and $\Phi(\mathbf{w})\in\Phi(C^{\perp})$, which is implies that $\Phi(\mathbf{w})\in\Phi(C)\cap\Phi(C^{\perp})$. Therefore,

[TABLE]

On the other hand, suppose that $\mathbf{v}\in\Phi(C)\cap\Phi(C^{\perp})$, then there exist $\mathbf{x}\in C$ and $\mathbf{y}\in C^{\perp}$ such that $\mathbf{v}=\Phi(\mathbf{x})=\Phi(\mathbf{y})$. Since $\Phi$ is an isomorphism, we have $\mathbf{x}=\mathbf{y}\in C\cap C^{\perp}$. Thus, $\mathbf{v}\in\Phi(C\cap C^{\perp})$. This means that

[TABLE]

Combining $(3.1)$ and $(3.2)$, we get the desired equality.

Now, we assume that a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ is $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD, then by $\Phi(C\cap C^{\perp})=\Phi(C)\cap\Phi(C^{\perp})$ and $\Phi(C^{\perp})=\Phi(C)^{\perp}$, we know that $\Phi(C)$ is a binary LCD code over $\mathbb{Z}_{2}$.

Conversely, let $\Phi(C)$ be is a binary LCD code. Assume that $C$ is not a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code, then there exists $0\neq\mathbf{x}\in C\cap C^{\perp}$ such that

[TABLE]

where $\mathbf{x}=(a_{0},\ldots,a_{s-1},b_{0}+uc_{0},\ldots,b_{t-1}+uc_{t-1})$.

We assert that $\Phi(\mathbf{x})\neq 0$. Otherwise, by $\Phi(\mathbf{x})=0$, we have

[TABLE]

and

[TABLE]

This means that $\mathbf{x}=0$. This is a contradiction.

Therefore, $0\neq\Phi(\mathbf{x})\in\Phi(C)\cap\Phi(C^{\perp})$, which is a contradiction to the assumption. ∎

4 Examples of new binary LCD codes

In this section, examples of some new binary codes derived from this family of $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD codes as their Gray images are presented.

Example 3**.**

Let $C$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code of type $(9,9)$ with the generator matrix

[TABLE]

Obviously,

[TABLE]

Thus, $det(GG^{T})=1$. By Theorem 3.5, the $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ is a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. Again by Theorem 3.12, the binary image of this code is a binary LCD code with parameters $[27,8,10]_{2}$. This is an optimal code which is obtained directly in contrast to the indirect constructions presented in [10].

Example 4**.**

Let $C$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code of type $(7,7)$ with the generator matrix

[TABLE]

Obviously, $det(GG^{T})=1$. By Theorem 3.5, the $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ is an $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. Again by Theorem 3.12, the binary image of this code is a binary LCD code with parameters $[21,8,3]_{2}$.

Example 5**.**

Let $C$ be a $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code of type $(31,31)$ with the generator matrix

[TABLE]

Obviously,

[TABLE]

Thus, $det(GG^{T})=1$. By Theorem 3.5, the $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear code $C$ is an $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD code. Again by Theorem 3.12, the binary image of this code is an optimal binary LCD code with parameters $[93,7,46]_{2}$.

Example 6**.**

Combining Corollary 3.7 and Theorem 3.12, we obtain some new binary LCD codes in Table 1.

5 Conclusion

We have developed new methods of constructing binary LCD codes from $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-codes. Using those methods, we have constructed good binary LCD codes. We believe that $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD codes will be a good source for constructing good binary LCD codes. In a future work, in order to construct new binary LCD, we will use the computer algebra system MAGMA to find more good $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-LCD codes.